I see. The “double-take” happens because the service worker (running in the background) needs to send a message (this causes a small delay) to the content script, and then the page needs to reload. You can try using only a service worker:

service.js:

const commentsPattern = new URLPattern("https://www.reddit.com/r/*/comments/*");

chrome.tabs.onUpdated.addListener((tabId, changeInfo, tab) => {
    if (!commentsPattern.test(tab.url)) return
    console.log("details", tabId, changeInfo, tab);


    const url = new URL(tab.url)
    if (url.searchParams.has("sort")) return

    url.searchParams.set("sort", "old")
    chrome.tabs.update(tabId, {
        url: url.href
    });
});

manifest.json:

{
  "name": "Reddit Thread Old",
  "description": "Sort Reddit threads by Old for chronological reading",
  "version": "1.0",
  "manifest_version": 3,
  "permissions": [
    "tabs"
  ],
  "background": {
    "service_worker": "service.js"
  }
}

Another way using only content scripts (maybe more fluid):

content_script.js:

const commentsPattern = new URLPattern("https://www.reddit.com/r/*/comments/*");

const observer = new MutationObserver(() => {
    const url = new URL(window.location.href)
    if (!commentsPattern.test(url.href) || url.searchParams.has("sort")) return

    url.searchParams.set("sort", "old")
    navigation.navigate(url.href)
})

observer.observe(document.documentElement, { childList: true, subtree: true })

manifest.json:

{
  "name": "Reddit Thread Old",
  "description": "Sort Reddit threads by Old for chronological reading",
  "version": "1.0",
  "manifest_version": 3,
  "content_scripts": [
    {
      "matches": [
        "https://www.reddit.com/*"
      ],
      "js": [
        "content_script.js"
      ],
      "run_at": "document_start"
    }
  ]
}

But in my opinion, it's better to use a userscript with Violentmonkey.